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arXiv 2006.08227math.AP
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关于椭圆复形Navier-Stokes型方程的稳定性现象

On the stability phenomenon of the Navier-Stokes type Equations for Elliptic Complexes

Andrei Parfenov, Alexander Shlapunov

发表信息：
Complex Variables and Elliptic Equations, Volume 66, 2021 - Issue 6-7, Pages 1122-1150
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摘要：

设${\mathcal X}$是黎曼$n$维光滑紧闭流形，$n\geq 2$，$E^i$是$\mathcal X$上的光滑向量丛，$\{A^i,E^i\}$是一阶线性算子的椭圆微分复形.我们考虑的算子方程，诱导的Navier-Stokes型方程与$\{A^i,E^i\}$的各向异性的Hölder空间的尺度上的层${\mathcal X} \times [0,T]$与有限时间$T > 0$。利用该尺度空间上的微分$A^i$和抛物算子的性质，将方程化为形式为$(I+K) u = f$的非线性Fredholm算子方程，其中$K$是紧连续算子.证明了Fréchet导数$(I+K)'$在所考虑的每个Banach空间的每一点上连续可逆，并且映射$(I+K)$在该空间中是开的和单射的.

英文摘要：

Let ${\mathcal X}$ be a Riemannian $n$-dimensional smooth compact closed manifold, $n\geq 2$, $E^i$ be smooth vector bundles over $\mathcal X$ and $\{A^i,E^i\}$ be an elliptic differential complex of linear first order operators. We consider the operator equations, induced by the Navier-Stokes type equations associated with $\{A^i,E^i\}$ on the scale of anisotropic Hölder spaces over the layer ${\mathcal X} \times [0,T]$ with finite time $T > 0$. Using the properties of the differentials $A^i$ and parabolic operators over this scale of spaces, we reduce the equations to a nonlinear Fredholm operator equation of the form $(I+K) u = f$, where $K$ is a compact continuous operator. It appears that the Fréchet derivative $(I+K)'$ is continuously invertible at every point of each Banach space under the consideration and the map $(I+K)$ is open and injective in the space.

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